In this section we find that
variations of Hodge structure on \({M}_{g,1}\)  with monodromy which is
“big" in a suitable sense provide examples of flat vector bundles on curves
whose isomonodromic deformation to a nearby curve is never semistable. We then produce such variations of
Hodge structure via the Kodaira-Parshin trick.
This will be used to prove
theorem:counterexample and contradicts earlier claimed theorems of
Biswas, Heu, and Hurtubise
[1]},
[2]}, and
[3]},
as described further in remark:bhh-error.
